🎲 Sample Mean

Sample Mean#

The arithmetic average of a sample, used to estimate the population mean.

Important

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What it is#

The sample mean \(\bar{x}\) is the arithmetic average of a sample — the sum of the observations divided by their count. It is the statistic used to estimate the true population mean \(\mu\):

\[\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i.\]

Examples#

Test scores \(\{80, 85, 90, 95, 100\}\) give \(\bar{x} = 450/5 = 90\). Ten people with total height 1,720 cm give \(\bar{x} = 172\) cm.

Its three key properties#

Unbiased — \(\mathbb{E}[\bar{x}] = \mu\); on average the sample mean equals the population mean.
Sampling distribution (CLT) — for large \(n\), the Central Limit Theorem makes \(\bar{x}\) approximately normal,

\[\bar{x} \sim N\!\left(\mu, \frac{\sigma^2}{n}\right),\]

with standard error of the mean \(SE = \sigma/\sqrt{n}\) — so its variability shrinks as \(n\) grows.
Outlier-sensitive — being a sum, the mean is pulled by extreme values (unlike the median).

Where it shows up#

The sample mean is everywhere: descriptive summaries, estimating \(\mu\), hypothesis tests (the one-sample t-test), and confidence intervals. It is the best unbiased estimator of the population mean.

Theme: Probability & Statistics Foundations · All terminology

Hint

Mind map — connected ideas

Sample Standard Deviation · True Mean (Population Mean) · Standard Error (SE) · Parameter(s) of Interest · Frequentist · Regression Coefficient

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More in Probability & Statistics Foundations

Beta Distribution · Confidence Level · Correlation · Critical Value · Cumulative Distribution Function (CDF) · Frequentist · IID (Independent and Identically Distributed) · Likelihood · Margin of Error (MoE) · Mean · Median · Normal Distribution · Outlier · Population Proportion